Abstract

The aim of this paper is to decide whether a linear differential equation with polynomial coefficients depending on parameters has got polynomial solutions. More precisely we want to construct a finite set $T$ of necessary and sufficient algebraic and arithmetic conditions such that there is a polynomial solution if and only if the parameters belong to $T$. The presence of Diophantine equations makes the general problem undecidable. We get such a set $T$ when the recurrence relation associated to the equation (in an appropriate basis) has got two terms. Using hypergeometric sequences we also succeed in constructing sufficient conditions for a family of equations.

Abstract: